Background
For which integers N can the Jacobian J of a curve C
of genus 2 over Q contain a rational torsion point
p of order N?
Without further assumptions on C or J, there is no hope at present for bounding N from above, so the best we can do is seek examples with large N.
The problem naturally splits into two parts, according as J is simple or split. Here an abelian variety is said to be “simple” if and only if it is not isogenous to a product of abelian varieties of lower dimension, and “(completely) split” if it is isogenous to a product of elliptic curves; in our case J is two-dimensional, so it must be one or the other. Note that the elliptic curves are allowed to be defined over some extension of Q; thus we use “simple” to mean “absolutely simple”: not split over any number field.
In the non-simple (split) case, the state of the art is represented by the paper “Large torsion subgroups of split Jacobians of curves of genus two or three” by Everett Howe, Franck Leprévost, and Bjorn Poonen (Forum Math.12 (2000) #3, 315-364). There one finds examples of curves or families of (C,J,p,N) with J split for various values of N, the largest of which is 63. The constructions hinge on the torsion structure of the elliptic curves whose product is isogenous with J. These are controlled by various “modular curves”, whose study is an active and highly developed topic in number theory.
In the simple case, the theory of modular curves is no longer available. Torsion points of high order are usually constructed by forcing C to have two or three non-Weierstrass points that support several divisors linearly equivalent to zero. This method is described in pages 84-87 of J.W.S.Cassels and E.V.Flynn's book Prolegomena to a middlebrow arithmetic of curves of genus 2 (LMS Lecture Notes 230), Cambridge University Press, 1996. This method was used by Leprévost to construct rational N-torsion points for N as high as 29 (in his paper “Jacobiennes décomposables de certaines courbes de genre 2: torsion et simplicité”, J. Théorie des Nombres de Bordeaux 7 (1995) #1, 283-306) and 30 (in an unpublished 4/1996 preprint, reported by Everett Howe). While Leprévost found only a single curve with a 29-torsion divisor, he obtained an infinite (one-parameter) family of curves with N=30.
Announcement of new results
By systematically pursuing Leprévost's method
and natural variations of it,
I found several new and larger values of N:
a one-parameter family of curves whose Jacobian has a torsion point
of order 32; two curves each of whose Jacobians
has a torsion point of order 34; and one curve
each for 39 and 40.
The N=39 curve is also remarkable for having minimal automorphism group
(only the identity and the hyperelliptic involution)
and four pairs of non-Weierstrass points,
each of which differs from any Weierstrass point by a torsion divisor.
Without extra symmetry, parameter counts lead one to expect only three
such pairs, because the moduli space of genus-2 curves has dimension 3;
the N=39 curve is the only one known (as of 6/2002) with a fourth pair.
Explicit formulas
We exhibit the four curves with N=34, 34, 39, 40.
We refrain from giving an explicit formula for the N=32 family,
because the coefficients of the curves are very complicated
rational functions of the family's parameter;
we thus content ourselves with exhibiting a curve
in that family with reasonably small coefficients.
The simplicity of each of the curves' Jacobians was checked
by Leprévost's criterion (Lemme 3.1.2 of his paper):
a genus-2 curve C over Q has simple Jacobian
if there is a prime l of good reduction such that the Galois group
of the characteristic polynomial of Frobenius of the reduced curve
is the 8-element dihedral group D4. [The Galois group
is a priori contained in D4, because the roots
of the polynomial are four complex numbers of the same absolute value;
Leprévost shows that if J is isogenous to a product of elliptic
curves then the Galois group is strictly smaller than D4.]
Bjorn Poonen has a program that computes, for most genus-2 curves C, all points that differ from their hyperelliptic conjugates by torsion divisors; his paper that develops the requisite theory and describes the program is available online in pdf or compressed dvi. He confirmed that for the N=39 curve these four pairs of points, together with the six Weierstrass points, are the only points of C, even over the complex numbers, that differ from their hyperelliptic conjugates by torsion divisors.
Poonen also observed that J, while simple, appears to have real multiplication by Q(sqrt(2)), which would make J conjecturally modular; he reports using Qing Liu's program to determine the arithmetic conductor of J: it is 2452=7452, so J should be isogenous to a simple factor of the Jacobian J0(245) of the modular curve X0(245)=X0(725), and not to a factor of J0(M) for any M<245. Again as with N=39, good reduction at 2 can be confirmed by “uncompleting the square” mod 4 to obtain an equivalent formula for the curve, here
There are over a hundred examples over Q with five point pairs (both with and without a rational Weierstrass point), and many nonconstant families with four point pairs parametrized by P1. The above six-point curve was found by specializing the (q, 5q, 13q, 29q) family
has four points (x,y)=(inf,inf3), (0,1-t), (-1,1), and (1,-1) over C(t) giving q,2q,8q,13q, and the specialization t=5 has two further points, with x=-2 and x=-130/77, that yield divisors that differ from 4q and 38q by a 2-torsion point. A more complicated family of curves with a rational Weierstrass point as well as 6q,8q,9q,13q specializes to
The points on C(b) giving q,5q,13q,29q are (x,y)=(inf,inf3), (0,-(b+1)), (1,1), and (-1,2b+1). The choice of coefficients 1,5,13,29 may seem bizarre, but in fact {-29,-13,-5,-1,1,5,13,29} is the lexicogrphically first symmetric set of 2*4 odd integers satisfying the necessary condition that all nonzero sums of elements of the set be distinct. Likewise {-13,-8,-2,-1,1,2,8,13} is one of the first such sets if we drop the parity condition.
Michael Stoll also computed the canonical heights of q on C(b), and of its specialization to b=-5/2, finding 1/357 and .000797... As far as we know, .000797... is the smallest positive canonical height yet exhibited for a point on the Jacobian of a curve of genus 2 over Q. Over P1, the record is 1/546, belonging to the q,2q,8q,13q curve over Q(t) exhibited above.
In the more familiar case of genus 1 (elliptic curves), there are questions of similar flavor concerning points of low positive height; one can also consider integral points on the minimal (Néron) model of an elliptic curve as an analogue of rational points on a curve of genus 2. Some remarkable examples of nontorsion points with low height and/or many integral multiples are listed here.